A372844 a(n) is the number of parking functions of order n for which the fourth spot is lucky.

64, 708, 9421, 148992, 2742090, 57671104, 1365730231, 35980443648, 1044117402868, 33098695234560, 1138160856018369, 42200676331159552, 1678427133899138494, 71282668099352051712, 3219814814790580711915, 154137012617228775849984, 7795444201708762192584744, 415337944634097426474729472

Offset: 4

Author

Kimberly P. Hadaway, Jun 26 2024

Keywords

• nonn

Comments

A lucky spot is one which is parked in by a car which prefers that spot.

Examples

			For clarity, we write parentheses around parking functions. For n = 4, there are a(4) = 64 solutions. An example of a parking function of order 4 with a lucky fourth spot is (1,4,2,2); here, the second car parks in the fourth spot which is its preferred spot. This parking function contributes to our count. A non-example is the parking function (1,2,1,2); here, the last car parks in the fourth spot, but its preference is spot 2. This parking function does not contribute to our count.
		

Links

• Steve Butler, Kimberly Hadaway, Victoria Lenius, Preston Martens, and Marshall Moats, Lucky cars and lucky spots in parking functions, arXiv:2412.07873 [math.CO], 2024. See p. 10.

Crossrefs

Cf. A000272 (for first spot), A372842 (for second spot), A372843 (for third spot), and A372845 (for fifth spot).

Programs

Mathematica

a[n_]:=(5/8)*(n+1)^(n-1)-(1/8)*(13*n^2-26*n+9)*(n-3)^(n-3); Array[a,19,4] (* Stefano Spezia, Jun 26 2024 *)

Python

def A372844(n): return 5*(n+1)**(n-1)-(13*(n-1)**2-4)*(n-3)**(n-3)>>3 # Chai Wah Wu, Jun 26 2024

Formula

a(n) = (5/8)*(n+1)^(n-1) - (1/8)*(13*n^2 - 26*n + 9)*(n-3)^(n-3).